A SINGULAR CONTROL APPROACH TO HIGHLY DAMPED 2ND-ORDER ABSTRACT EQUATIONS AND APPLICATIONS

In this paper we restudy, by a radically different approach, the optim al quadratic cost problem for an abstract dynamics, which models a spe cial class of second-order partial differential equations subject to h igh internal damping and acted upon by boundary control. A theory for this problem was recently derived in [LLP] and [T1] (see also [T2]) by a change of variable method and by a direct approach, respectively. U nlike [LLP] and [T1], the approach of the present paper is based on si ngular control theory, combined with regularity theory of the optimal pair from [T1]. This way, not only do we rederive the basic control-th eoretic results of [LLP] and [T1]-the (first) synthesis of the optimal pair, and the (first) nonstandard algebraic Riccati equation for the (unique) Riccati operator which enters into the gain operator of the s ynthesis-but in addition, this method also yields new results-a second form of the synthesis of the optimal pair, and a second (still nonsta ndard) algebraic Riccati equation for the (still unique) Riccati opera tor of the synthesis. These results, which show new pathologies in the solution of the problem, are new even in the finite-dimensional case.